Theorem sbal1yz 2054

Description: Lemma for proving sbal1 2055. Same as sbal1 2055 but with an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)

Assertion

Ref	Expression
sbal1yz	⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal1yz

Step	Hyp	Ref	Expression
1		dveeq2or 1864	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧)
2		equcom 1754	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
3	2	nfbii 1521	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ Ⅎ𝑥 𝑧 = 𝑦)
4		19.21t 1630	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
5	3, 4	sylbi 121	. . . . . . 7 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
6	5	orim2i 768	. . . . . 6 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))))
7	1, 6	ax-mp 5	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
8	7	ori 730	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
9	8	albidv 1872	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑)))
10		alcom 1526	. . . 4 ⊢ (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑧 = 𝑦 → 𝜑))
11		sb6 1935	. . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑))
12	2	imbi1i 238	. . . . . . 7 ⊢ ((𝑦 = 𝑧 → 𝜑) ↔ (𝑧 = 𝑦 → 𝜑))
13	12	albii 1518	. . . . . 6 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → 𝜑))
14	11, 13	bitri 184	. . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑧 = 𝑦 → 𝜑))
15	14	albii 1518	. . . 4 ⊢ (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑧 = 𝑦 → 𝜑))
16	10, 15	bitr4i 187	. . 3 ⊢ (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
17		sb6 1935	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
18	2	imbi1i 238	. . . . 5 ⊢ ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))
19	18	albii 1518	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑))
20	17, 19	bitr2i 185	. . 3 ⊢ (∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑) ↔ [𝑧 / 𝑦]∀𝑥𝜑)
21	9, 16, 20	3bitr3g 222	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
22	21	bicomd 141	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 715  ∀wal 1395  Ⅎwnf 1508  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sbal1  2055